Georgia Institute of Technology College of Sciences

Discrete recurrent neural networks have a long history, starting with McCulloch and Pitt's seminal paper in 1943 that ushered in the computer era as well as inspired the modern field of deep learning. We discuss some of the mathematics underlying the special case of Hopfield networks, which have received considerable attention in recent years (Hopfield won the Nobel prize). Specifically, we show how these networks can robustly store an exponential number of memories, originally thought to be impossible using only linear-threshold computational primitives. Applications to error-correcting codes, generalization in AI, and graph isomorphism will be discussed.  

A fundamental question in contact topology is to understand which symplectic 4-manifolds can have a given contact 3-manifold as their boundary. Such 4-manifolds are called symplectic fillings, and their classification is an important problem in symplectic topology.In this talk, I will introduce the problem of classifying symplectic fillings and discuss how cutting a contact 3-manifold along certain special tori can help us understand its fillings. I will explain how this approach can bbe used totudy symplectic fillings of a large class of plumbed 3-manifolds, including examples arising from contact surgery diagrams. The talk will focus on the main ideas and examples rather than technical details.

This is joint work with Austin Christian.

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifold of $(\rho_0,v_0)$. The proof is based on the Lyapunov--Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Third of three talks.

In Bernoulli percolation, we place edges independently with probability p between vertices of a lattice, and study the resulting random geometry. In the first talk, I will give a friendly introduction to the model and discuss some of the motivating questions and the history of methods in the high dimensional setting. In the second talk, I will discuss our recently developed tool, the incipient infinite cluster, and describe how we use it to derive the exact limiting distribution of distances within clusters, as well as prove sharp asymptotics for k-point connectivity functions. These results together establish several aspects of the conjecture that the scaling limit of high dimensional critical clusters is the Integrated Super-Brownian Excursion. Based on joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe.

Abstract: Lov\'{a}sz proved that two graphs $G$ and $H$ are isomorphic if they have the same homomorphism counts from all graphs, and Dvo\v{r}\'{a}k showed that it suffices to count homomorphisms from all $2$-degenerate graphs. Homomorphism counts from various other graph classes have been shown to give interesting graph relations weaker than isomorphism. Many such classes are minor-closed, and Roberson conjectured that homomorphism indistinguishability over any proper minor-closed class $\mathcal{M}$ is a strictly weaker relation than isomorphism; that is, there exist graphs $G$ and $H$ that are non-isomorphic but satisfy $\hom(K,G)=\hom(K,H)$ for all $K\in\mathcal{M}$. We prove an analogue of this conjecture for immersions. Joint work with Andrea Jim\'enez, Benjamin Moore, and Daniel A. Quiroz. 

Finding a path between two probability distributions is a recurring problem in optimal transport, generative modeling, population dynamics, and control. A broad class of these problems can be formulated through an action functional defined over curves of probability distributions. The corresponding first-order optimality conditions lead to a coupled Hamilton–Jacobi and continuity system, which can be written as a boundary-value Hamiltonian flow in density space.

Nash's isometric embedding theorem is a fundamental result in differential geometry, establishing that every smooth Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. Nash's original proof relies on a sophisticated iterative scheme to overcome the loss of derivatives arising from the nonlinear isometric embedding equations. In the late 1980s, Matthias Günther introduced an alternative approach that avoids the Nash–Moser iteration scheme and the associated tame estimates, considerably simplifying the argument. In this talk, we will introduce the isometric embedding problem, discuss the main difficulties in solving the underlying nonlinear system of partial differential equations, and outline the key ideas behind Günther's approach. The talk will be expository, emphasizing the geometric intuition and analytical techniques underlying the proof.

Parry proved in 1964 that a mixing subshift of finite type has a unique measure of maximal entropy. Upper semi-continuity of entropy implies that every invariant measure with nearly maximal entropy must be close to the MME in the Wasserstein metric, which metrizes the weak* topology. In 2015, Kadyrov proved an effective uniqueness result that provides a quantitative version of this statement.

I will present joint work with Teena Kumari, in which we prove that Kadyrov's result remains true when the Wasserstein metric is replaced with Ornstein's d-bar metric. This strengthens the qualitative result from Ornstein theory that the MME is finitely determined since it is isomorphic to a Bernoulli shift. I will also discuss connections to recent work by a number of authors on derivatives of the pressure function and Gaussian concentration bounds.

After centering and scaling, the length of the longest weakly increasing subsequences of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting Robinson–Schensted–Knuth (RSK) Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the

limiting Brownian functional may fail to have the law of the largest eigenvalue of {\it any} centered jointly Gaussian $3\times 3$ Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.

Joint work with Dorian Kidonakis

In many applications of mathematical optimization, one may wish to optimize an objective function without access to its derivatives (gradients). These situations call for derivative-free optimization (DFO) methods. Among the most successful approaches in practice are model-based trust-region methods, such as those pioneered by M.J.D Powell. These methods rely on function approximations via low degree polynomials and carefully adapt the local geometry of interpolation points to balance exploration and exploitation.  While relatively complex to implement, these methods are now available in standard scientific computing platforms, including MATLAB and SciPy. However, theoretical analysis of their computational complexity lagged behind practice. In particular, it is important to bound the number of function evaluations required to achieve a desired level of accuracy. Using a mix of handy concepts from approximation, sample set geometry, Lagrangian interpolation and linear algebra we systematically derive complexity bounds for classical model-based trust-region methods and their modern variations. We establish, for the first time, that these methods can have the same worst case complexity than any other known DFO method.